Involution-symmetry conjecture for master-function critical points

Let $\boldsymbol{z}$ be a generic configuration on the unit circle, and let $\Phi_{m,n}(\boldsymbol{z},\boldsymbol{\xi})$ be the master function. Involution-symmetry conjecture. Every critical point $\boldsymbol{\xi}$ of $\Phi_{m,n}(\boldsymbol{z},\boldsymbol{\xi})$ is invariant under the relevant involution. This is proposed as an equivalent formulation of the quadratic-differential classification conjectures, and no proof is supplied.